Changing character of electronic transitions in graphene: From single-particle excitations to plasmons

In this paper we clarify the nature of $\pi$ and $\pi +\sigma$ electron excitations in pristine graphene. We clearly demonstrate the continuous transition from single particle to collective character of such excitations and how screening modifies their dispersion relations. We prove that $\pi$ and $\pi +\sigma$ plasmons do exist in graphene, though occurring only for a particular range of wave vectors and with finite damping rate. Particular attention is paid to comparing the theoretical results with available EELS measurements in optical $\left(\mathrm{Q}\approx 0\right)$ and other $\left(\mathrm{Q}\ne 0\right)$ limits. The conclusions, based on microscopic numerical results, are confirmed in an approximate analytical approach.

Figure 1

Main panel: graphene band structure along the ${\mathrm{M}}^{\prime }\text{−}\mathrm{M}$ (black lines) and $\mathrm{M}\text{−}\Gamma$ (blue lines) directions of the BZ. Interband transitions between $\pi$ bands are marked with thick arrows, while the $\sigma$ band transitions are marked with thin arrows. Inset: BZ of graphene with black and blue arrows showing directions of corresponding band structures in the main panel.

Figure 2

Real (green) and imaginary (blue) parts of the macroscopic dielectric function ${\mathcal{E}}_{\mathrm{M}}(\mathrm{Q},\omega )$, dynamical screening factor $\mathrm{D}(\mathrm{Q},\omega )$ (light blue), and spectra of electronic excitations (black) in pristine graphene for six different Q vectors along the $\Gamma \mathrm{M}$ direction: (a) ${\mathrm{Q}}_{\Gamma \mathrm{M}}=0.008\mathrm{a}.\mathrm{u}.$, (b) ${\mathrm{Q}}_{\Gamma \mathrm{M}}=0.039\mathrm{a}.\mathrm{u}.$, (c) ${\mathrm{Q}}_{\Gamma \mathrm{M}}=0.078\mathrm{a}.\mathrm{u}.$, (d) ${\mathrm{Q}}_{\Gamma \mathrm{M}}=0.147\mathrm{a}.\mathrm{u}.$, (e) ${\mathrm{Q}}_{\Gamma \mathrm{M}}=0.225\mathrm{a}.\mathrm{u}.$, and (f) ${\mathrm{Q}}_{\Gamma \mathrm{M}}=0.303\mathrm{a}.\mathrm{u}.$ Red vertical dashed lines denote the energy positions of $\pi ,{\sigma }_1$, and ${\sigma }_2$ plasmons for each Q vector.

Figure 3

(a) Energies of $\pi \to {\pi }^{*}$ interband transitions (blue) and the $\pi$ plasmon (red) as functions of the wave vector Q along $\Gamma \mathrm{M}$ direction obtained from DFT-RPA method. Black unfilled diamonds represent the experimental data from Ref. [12]. (b) Same as in (a) for $\sigma \to {\sigma }_1^{*}$ and $\sigma \to {\sigma }_2^{*}$ (blue) transitions and ${\sigma }_1$ and ${\sigma }_2$ plasmons (red). The point $\mathrm{Q}=0$ is treated separately [31]. (c) Same as in (a) but showing the results obtained with the TB-RPA method. In addition the points where ${\varepsilon }_1(\mathrm{Q},\omega )=0$ are shown by the black dashed line and the corresponding $\sqrt{\mathrm{Q}}$ fit with the orange line. Inset: real (green) and imaginary (dashed blue) parts of $\varepsilon (\mathrm{Q},\omega )$ and $\mathrm{S}(\mathrm{Q},\omega )$ (black) for $\mathrm{Q}=0.01\mathrm{a}.\mathrm{u}.$ as obtained with TB-RPA method.

Figure 4

Real part of the dielectric function for a several Q wave vectors. Blue line represents the results obtained by DFT-RPA method with LFE included $\left(\mathrm{Re}{\mathcal{E}}_{\mathrm{M}}\right)$, red dashed-dotted line without LFE $\left(\mathrm{Re}{\mathcal{E}}_{00}\right)$, and the green line represents the TB-RPA results $\left(\mathrm{Re}\varepsilon \right)$. Zero values of $\mathrm{Re}\mathcal{E}$ are denoted by black horizontal lines.